From curvature to Kovacic: a geometric approach to integrability of scalar ODEs
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| Format: | Preprint |
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2026
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| author | Pan-Collantes, A. J. Álvarez-García, J. A. |
| author_facet | Pan-Collantes, A. J. Álvarez-García, J. A. |
| contents | We study first-order ordinary differential equations such that the intrinsic Gauss curvature of the associated surface depends only on the independent variable: $\mathcal{K}(x,u)=κ(x)$, showing that this geometrically motivated class of equations admits a threefold connection to the second-order linear operator $L=d^2/dx^2+κ(x)$: the divergence along every solution satisfies a Riccati equation that linearizes to $L(y)=0$; every solution of the first-order equation satisfies the non-homogeneous equation $L(u)=c(x)$; and solutions of $L(y)=0$ give rise to integrating factors for the original nonlinear equation. By means of differential Galois theory, we prove that the nonlinear equation is integrable by quadratures if and only if $L$ admits a non-zero Liouvillian solution; when $κ$ is rational, Kovacic's algorithm provides a complete decision procedure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_05752 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From curvature to Kovacic: a geometric approach to integrability of scalar ODEs Pan-Collantes, A. J. Álvarez-García, J. A. Classical Analysis and ODEs Exactly Solvable and Integrable Systems We study first-order ordinary differential equations such that the intrinsic Gauss curvature of the associated surface depends only on the independent variable: $\mathcal{K}(x,u)=κ(x)$, showing that this geometrically motivated class of equations admits a threefold connection to the second-order linear operator $L=d^2/dx^2+κ(x)$: the divergence along every solution satisfies a Riccati equation that linearizes to $L(y)=0$; every solution of the first-order equation satisfies the non-homogeneous equation $L(u)=c(x)$; and solutions of $L(y)=0$ give rise to integrating factors for the original nonlinear equation. By means of differential Galois theory, we prove that the nonlinear equation is integrable by quadratures if and only if $L$ admits a non-zero Liouvillian solution; when $κ$ is rational, Kovacic's algorithm provides a complete decision procedure. |
| title | From curvature to Kovacic: a geometric approach to integrability of scalar ODEs |
| topic | Classical Analysis and ODEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2604.05752 |